Journal of Theoretical Probability

, Volume 23, Issue 3, pp 888–903

# Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem

• Dalibor Volný
Article

## Abstract

Let (X i ) be a stationary and ergodic Markov chain with kernel Q and f an L 2 function on its state space. If Q is a normal operator and f=(IQ)1/2 g (which is equivalent to the convergence of $$\sum_{n=1}^{\infty}\frac{\sum_{k=0}^{n-1}Q^{k}f}{n^{3/2}}$$ in L 2), we have the central limit theorem [cf. (Derriennic and Lin in C.R. Acad. Sci. Paris, Sér. I 323:1053–1057, 1996; Gordin and Lifšic in Third Vilnius conference on probability and statistics, vol. 1, pp. 147–148, 1981)]. Without assuming normality of Q, the CLT is implied by the convergence of $$\sum_{n=1}^{\infty}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}$$ , in particular by $$\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=o(\sqrt{n}/\log^{q}n)$$ , q>1 by Maxwell and Woodroofe (Ann. Probab. 28:713–724, 2000) and Wu and Woodroofe (Ann. Probab. 32:1674–1690, 2004), respectively. We show that if Q is not normal and f∈(IQ)1/2 L 2, or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to $$\sum_{n=1}^{\infty}c_{n}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}<\infty$$ for some sequence c n ↘0, or by $$\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=O(\sqrt{n}/\log n)$$ , the CLT need not hold.

## Keywords

Martingale approximation Martingale difference sequence Strictly stationary process Markov chain Central limit theorem

## Mathematics Subject Classification (2000)

60G10 60G42 28D05 60F05

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